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RC Circuits

RC Circuits. circuits in which the currents vary in time rate of charging a cap depends on C and R of circuit differential equations.

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RC Circuits

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  1. RC Circuits • circuits in which the currents vary in time • rate of charging a cap depends on C and R of circuit • differential equations

  2. Quiz:After the switch is closed, the light from the bulb:A) Is brightest just after the switch is closed, then fades slowly and disappears.B) Increases gradually to maximum brightness, over a few seconds, then remains steady.C) Comes to maximum brightness immediately, and remains steady.D) The bulb remains off.

  3. I Given: R, C, qo (initial charge) Find: q(t) and I(t) when switch is closed q C R -q 1) (Kirchhoff’s Loop Rule) 2) Discharging a Capacitor (- sign because q decreases for I > 0 That is, current in circuit equals the decrease of charge on the capacitor)

  4. Combine 1) and 2) to get: I where: q = q(t) q(0) = qo q C R -q This is a differential equation for the function q(t), subject to the initial condition q(0) = q0 . We are looking for a function which is proportional to its own first derivative (since dq/dt ~ -q).

  5. Solution: RC is called the “time constant” or “characteristic time” of the circuit. Units: 1 Ω x 1 F = 1 second (show this!) Write t (“tau”) = RC, then: (discharging)

  6. q Discharging qo t 2 3 t = , q ≈ 0.37 qo = (qo/e) t = 2 , q ≈ 0.14 qo = (qo/e2) t = 3 , q ≈ 0.05 qo = (qo/e3) t  ∞ , q  0 = (qo/e∞)

  7. Draw a graph for I(t).

  8. Quiz • A capacitor is charged up to 18 volts, and then connected across a resistor. After 10 seconds, the capacitor voltage has fallen to 12 volts. • Find the time constant RC, and… • What will the voltage be after another 10 seconds (20 seconds total)? • 8V • 6V • 4V • 0

  9. Then, where t =RC. Charging a capacitor C is initially uncharged, and the switch is closed at t=0. After a long time, the capacitor has charge Qf . C R Question: What is Qf equal to?

  10. Charging a capacitor q Qf t 2 3 t = 0, q=0 t = 3 RC, q 0.95 Qf t = RC, q 0.63 Qf etc. t = 2 RC, q 0.86 Qf

  11. Draw a graph of I(t). Why is I=+dq/dt this time?

  12. Example 2 100 kΩ The capacitor is initially uncharged. After the switch is closed, find: 12 V 2 µF • Initial current • Initial voltage across the resistor • Initial voltage across the capacitor • Time for voltage across C to reach 0.63*12V • Final voltage across the resistor • Final voltage across the capacitor

  13. Solution

  14. Example:A 2kΩ and a 3kΩ resistors connected in parallel are connected in series with a 2uF and a 3uF capacitors that are connected in parallel. The power source is 120V. Find the charge on each capacitor as a function of time.

  15. Quiz:In a simple circuit with a capacitor, resistor and a switch, long time after the switch is closed, the current in the circuit will be:A) ε/RB) ε/τRC) ε/eRD) zero

  16. “RC” Circuits • a capacitor takes time to charge or discharge through a resistor • “time constant” or “characteristic time” • = RC • (1 ohm) x (1 farad) = 1 second

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